The difference, by the numbers
Both have lime juice. Only the IBA Tiki has aged rum, amaretto, hazelnut liqueur, Maraschino liqueur, passion fruit purée, pineapple juice and fresh ginger; only the Missionary’s Downfall has white rum, peach liqueur, honey syrup, mint and fresh pineapple. In parts, an IBA Tiki is 2 rum, ½ amaretto, 1 passion fruit, 3 pineapple and 1 lime; a Missionary’s Downfall is 1 rum, ½ peach, ½ lime, 1 honey and 2¾ fresh pineapple. By volume, the IBA Tiki is 52% mixer and juice, 26% spirit, 13% citrus and 9% liqueur; the Missionary’s Downfall is 49% mixer and juice, 17% spirit, 17% syrup and sugar, 9% liqueur and 8% citrus. That makes them close cousins on the ratio map.
The Missionary’s Downfall is the stronger drink in the glass (10.8% ABV against 8.7%), but the IBA Tiki is the bigger drink, so it holds more alcohol (1.65 standard drinks against 0.88). The IBA Tiki has more calories (about 263 against 196), but the Missionary’s Downfall has more sugar (26.7 g against 20.6 g). The IBA Tiki is shaken and served in a tiki mug; the Missionary’s Downfall is blended and served in a goblet.